Cac ham so so hoc

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  • i Hc Thi NguynTrng i Hc Khoa Hc

    Cao Sn

    CC HM S HC V NG DNG

    Chuyn ngnh: PHNG PHP TON S CP

    M S: 60.46.40

    LUN VN THC S TON HC

    Ngi hng dn khoa hc: GS.TSKH. H HUY KHOI

    Thi Nguyn - 2011

  • Cng trnh c hon thnh tiTrng i Hc Khoa Hc - i Hc Thi Nguyn

    Ngi hng dn khoa hc: GS.TSKH. H HUY KHOI

    Phn bin 1: PGS.TS. L Th Thanh Nhn

    Phn bin 2: TS. Nguyn Vn Ngc

    Lun vn c bo v trc hi ng chm lun vn hp ti:Trng i Hc Khoa Hc - i Hc Thi Nguyn

    Ngy 09 thng 09 nm 2011

    C th tm hiu tiTh Vin i Hc Thi Nguyn

  • 1Mc lc

    Mc lc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    M u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1 Cc hm s hc c bn 5

    1.1. Phi - hm -le . . . . . . . . . . . . . . . . . . . . . . . 5

    1.1.1. nh ngha . . . . . . . . . . . . . . . . . . . . . 5

    1.1.2. Cc tnh cht . . . . . . . . . . . . . . . . . . . . 6

    1.2. Hm tng cc c s dng ca n . . . . . . . . . . . . . 9

    1.2.1. nh ngha . . . . . . . . . . . . . . . . . . . . . 9

    1.2.2. Cc tnh cht . . . . . . . . . . . . . . . . . . . . 10

    1.3. Hm tng cc ch s ca s t nhin n . . . . . . . . . . 12

    1.3.1. nh ngha . . . . . . . . . . . . . . . . . . . . . 12

    1.3.2. Cc tnh cht . . . . . . . . . . . . . . . . . . . . 12

    1.4. Hm s cc c (n) . . . . . . . . . . . . . . . . . . . . 15

    1.4.1. nh ngha . . . . . . . . . . . . . . . . . . . . . 15

    1.4.2. Cc tnh cht . . . . . . . . . . . . . . . . . . . . 15

    1.5. Hm phn nguyn [x] . . . . . . . . . . . . . . . . . . . . 16

    1.5.1. nh ngha . . . . . . . . . . . . . . . . . . . . . 16

    1.5.2. Cc tnh cht . . . . . . . . . . . . . . . . . . . . 16

    2 Mt s ng dng ca cc hm s hc 18

    2.1. ng dng ca Phi - hm -le . . . . . . . . . . . . . . . 18

    2.1.1. Xt ng d mul ca mt s nguyn t . . . . 18

    2.1.2. Chng minh php chia vi d . . . . . . . . . . . 19

    2.1.3. Gii phng trnh ng d . . . . . . . . . . . . . 20

    2.1.4. Tm nghim nguyn ca phng trnh . . . . . . 21

  • 22.1.5. Tm cp ca s nguyn . . . . . . . . . . . . . . . 22

    2.1.6. Tm s t nhin tha mn tnh cht hm s (n) 23

    2.2. ng dng ca hm tng cc c s dng ca s t nhin n 24

    2.2.1. Chng minh mt s l hp s . . . . . . . . . . . 24

    2.2.2. Chng minh mt s l s hon ho . . . . . . . . 25

    2.2.3. Chng minh bt ng thc lin quan ti (n) . . 29

    2.3. ng dng ca hm S(n) . . . . . . . . . . . . . . . . . . 32

    2.3.1. Tm n bi S(n) tha mn mt h thc cho trc . 32

    2.3.2. Tnh gi tr S(n) . . . . . . . . . . . . . . . . . . 35

    2.3.3. Chng minh mt s biu thc lin quan ti S(n) . 37

    2.3.4. Xt tnh b chn ca hm s cha S(n) . . . . . . 39

    2.4. ng dng ca hm s cc c (n) . . . . . . . . . . . . 40

    2.4.1. Tm n tha mn mt iu kin cho trc ca (n) 40

    2.4.2. Mt s bt ng thc lin quan ti hm (n) . . 43

    2.4.3. Tm s nghim ca phng trnh bng phng

    php s dng (n) . . . . . . . . . . . . . . . . . 45

    2.5. ng dng ca hm phn nguyn [x] . . . . . . . . . . . . 46

    2.5.1. Bi ton nh tnh . . . . . . . . . . . . . . . . . 46

    2.5.2. Bi ton nh lng . . . . . . . . . . . . . . . . . 50

    Kt lun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    Ti liu tham kho . . . . . . . . . . . . . . . . . . . . . . . 55

  • 3M u

    S hc l mt trong nhng lnh vc c xa nht ca Ton hc, v

    cng l lnh vc tn ti nhiu nht nhng bi ton, nhng gi thuyt

    cha c cu tr li. Trn con ng tm kim li gii cho nhng gi

    thuyt , c nhiu t tng ln, nhiu l thuyt ln ca ton hc

    ny sinh. Hn na, trong nhng nm gn y, S hc khng ch l mt

    lnh vc ca ton hc l thuyt, m cn l lnh vc c nhiu ng dng,

    c bit trong lnh vc bo mt thng tin. V th, vic trang b nhng

    kin thc c bn v s hc ngay t trng ph thng l ht sc cn

    thit. Khng nh nhiu ngnh khc ca ton hc, c rt nhiu thnh

    tu hin i v quan trng ca S hc c th hiu c ch vi nhng

    kin thc ph thng c nng cao mt bc. Do , y chnh l lnh

    vc thun li a hc sinh tip cn nhanh vi khoa hc hin i. Tuy

    nhin, trong chng trnh S hc trng ph thng hin nay, mn S

    hc cha c ginh nhiu thi gian. Cng v th m hc sinh thng

    rt lng tng khi gii bi ton S hc, c bit l trong cc k thi chn

    hc sinh gii.

    Trong phn S hc, cc hm s hc ng vai tr quan trng trong

    vic hnh thnh v nghin cu l thuyt hon thin. y l mt vn

    c in v quan trng ca S hc. Cc bi tp ng dng cc hm s

    hc c bn c cp nhiu trong cc k thi chn hc sinh gii cp

    tnh (thnh ph), Quc gia, Quc t.

    Mc ch chnh ca lun vn l nu ra c mt s ng dng c bn

    ca cc hm s hc c bn (Phi-hm -le, hm tng cc c dng ca

    n, s cc c dng ca n, tng cc ch s ca s t nhin n, hm phn

    nguyn). C th l phn loi c cc dng bi tp ca cc hm s hc

    thng qua h thng bi tp s dng cc hm s hc v cc nh l c

  • 4bn ca S hc.

    Ni dung ca lun vn gm 2 chng

    Chng 1: Trnh by cc kin thc c bn ca cc hm s hc.

    Chng 2: Mt s ng dng ca cc hm s hc.

    Lun vn ny c hon thnh vi s hng dn v ch bo tn tnh

    ca GS.TSKH. H Huy Khoi - Vin Ton Hc H Ni. Thy dnh

    nhiu thi gian hng dn v gii p cc thc mc ca ti trong sut

    qu trnh lm lun vn. Ti xin c by t lng bit n su sc n

    Thy.

    Ti xin cm n ti S Ni V, S Gio dc v o to tnh Bc Ninh,

    trng THPT Thun Thnh 1, t Ton trng THPT Thun Thnh 1

    to iu kin gip ti hon thnh kha hc ny.

    Ti xin gi ti cc Thy C khoa Ton, phng o to sau i hc

    Trng i Hc Khoa Hc - i Hc Thi Nguyn, cng nh cc Thy

    c tham gia ging dy kha Cao hc 2009-2011 li cm n su sc v

    cng lao dy d trong sut qu trnh gio dc, o to ca nh trng.

    ng thi ti xin gi li cm n ti tp th lp Cao Hc Ton K3A

    Trng i Hc Khoa Hc ng vin gip ti trong qu trnh hc

    tp v lm lun vn ny.

    Tuy nhin do s hiu bit ca bn thn v khun kh ca lun vn

    thc s, nn chc rng trong qu trnh nghin cu khng trnh khi

    nhng thiu st, ti rt mong c s ng gp kin ca cc Thy C

    v c gi quan tm ti lun vn ny.

    Thi Nguyn, ngy 31 thng 07 nm 2011

    Tc gi

    Cao Sn

  • 5Chng 1

    Cc hm s hc c bn

    1.1. Phi - hm -le

    1.1.1. nh ngha

    nh ngha 1.1. Gi s n l mt s nguyn dng. Phi-hm -le ca

    n l s cc s nguyn dng khng vt qu n v nguyn t cng nhau

    vi n.

    K hiu Phi-hm -le l (n).

    V d 1.1. (1) = 1, (2) = 1, (3) = 2, (4) = 2, (5) = 4.

    nh ngha 1.2. Cho n l s nguyn dng. Nu a l s nguyn vi

    (a, n) = 1 th lun tn ti s nguyn dng k ak 1(mod n).S nguyn dng k b nht tha mn ak 1(mod n) c gi lcp ca s nguyn a (modn).

    nh ngha 1.3. Mt h thng d thu gn mul n l mt tp hp

    gm (n) s nguyn sao cho mi phn t ca tp hp u nguyn t

    cng nhau vi n v khng c hai phn t khc nhau no ng d mul

    n.

    V d 1.2. Tp hp {1, 3, 5, 7} l mt h thng d thu gn mul 8.Tp hp {3,1, 1, 3} cng vy.nh ngha 1.4. Mt tp hp A no c gi l mt h thng d y

    (mod n) nu vi bt k s x Z tn ti mt a A x a(modn).

  • 6V d 1.3. A = {0, 1, 2, ..., n 1} l mt h thng d y theomul n.

    Ch 1.1. D thy mt tp A = {a1, a2, ..., an} gm n s s l mt hthng d y theo mul n khi v ch khi ai = aj(modn) (ta k hiu"khng ng d" l =) vi i 6= j v i, j {1, 2, ..., n}.

    1.1.2. Cc tnh cht

    Tnh cht 1 . Gi s{r1, r2, ..., r(n)

    }l mt h thng d thu gn mul

    n, a l s nguyn dng v (a, n) = 1. Khi , tp hp{ar1, ar2, ..., ar(n)

    }cng l h thng d thu gn mul n.

    Chng minh. Trc tin ta chng t rng, mi s nguyn arj l nguyn

    t cng nhau vi n. Gi s ngc li, (arj, n) > 1 vi j no . Khi

    tn ti c nguyn t p ca (arj, n). Do , hoc p |a , hoc p |rj , tcl hoc p |a v p |n , hoc p |rj v p |n . Tuy nhin, khng th c p |rj vp |n v rj v n l nguyn t cng nhau. Tng t, khng th c p |a vp |n . Vy, arj v n nguyn t cng nhau vi mi j = 1, 2, ..., (n).

    Cn phi chng t hai s arj, ark (j 6= k) ty khng ng d muln. Gi s arj ark(mod n), j 6= k v 1 j (n) ; 1 k (n). V(a, n) = 1 nn ta suy ra rj rk(mod n). iu ny mu thun v rj, rkcng thuc mt h thng d thu gn ban u mul n.

    V d 1.4. Tp hp {1, 3, 5, 7} l mt h thng d thu gn mul 8.Do (3, 8) = 1 nn {3, 9, 15, 21} cng l mt h thng d mul 8.Tnh cht 2.(nh l -le) Gi s m l s nguyn dng v a l s

    nguyn vi (a,m) = 1. Khi a(m) 1 (modm).Chng minh. Gi s

    {r1, r2, ..., r(n)

    }l mt h thng thu gn gm

    cc s nguyn dng khng vt qu m v nguyn t cng nhau vi m.

    Do Tnh cht 1 v do (a,m) = 1, tp hp{ar1, ar2, ..., ar(n)

    }cng l

    mt h thng d thu gn mul m. Nh vy, cc thng d dng b

    nht ca ar1, ar2, ..., ar(m) phi l cc s nguyn r1, r2, ..., r(m) xp

    theo th t no . V th, nu ta nhn cc v t trong h thng d thu

    gn trn y, ta c: ar1.ar2...ar(m) r1.r2...r(m)(modm).

  • 7Do , a(m)r1r2...r(m) r1r2...r(m) (modm).V(r1, r2, ...r(m),m

    )= 1 nn a(m) 1 (modm).

    Ta c th tm nghch o mul n bng cch s dng nh l -le. Gi

    s a,m l cc s nguyn t cng nhau, khi :

    a.a(m)1 = a(m) 1 (modm).Vy a(m)1 l nghch o ca a mul m.

    V d 1.5. 2(9)1 = 261 = 25 = 32 5 (mod 9) l mt nghch o ca2 mul 9.

    H qu 1.1. (a, b) = 1 th a(b) + b(a) 1(mod ab).H qu 1.2. Vi (a, b) = 1 v n, v l hai s nguyn dng no th

    an(b) + bv(a) 1 (mod ab).H qu 1.3. Gi s c k (k 2) s nguyn dng m1,m2, ...,mk vchng nguyn t vi nhau tng i mt. t M = m1.m2...mk = mi.tivi i = 1, 2, ..., k ta c:

    tn1 + tn2 + ...+ t

    nk (t1 + t2 + ...+ tk)n(modM) vi n nguyn dng.

    By gi ta s cho cng thc tnh gi tr ca phi-hm -le

    ti n khi bit phn tch ca n ra tha s nguyn t.

    Tnh cht 3. Vi s nguyn t p ta c (p) = p 1. Ngc li, nu pl s nguyn dng sao cho (p) = p 1 th p l s nguyn t.Chng minh. Nu p l s nguyn t th vi mi s nguyn dng nh

    hn p u nguyn t cng nhau vi p. Do c p 1 s nguyn dng nhvy nn (p) = p 1.

    Ngc li, nu p l hp s th p c cc c d, 1 < d < p. Tt nhin

    p v d khng nguyn t cng nhau. Nh vy, trong cc s 1, 2, ..., p 1phi c nhng s khng nguyn t cng nhau vi p, nn (p) p 2.Theo gi thit, (p) = p 1. Vy p l s nguyn t.Tnh cht 4. Gi s p l s nguyn t v a l s nguyn dng. Khi :